Average Error: 25.9 → 24.6
Time: 3.2s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} = -\infty:\\ \;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 1.4420118311925315 \cdot 10^{300}:\\ \;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} = -\infty:\\
\;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 1.4420118311925315 \cdot 10^{300}:\\
\;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1 \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r116849 = a;
        double r116850 = c;
        double r116851 = r116849 * r116850;
        double r116852 = b;
        double r116853 = d;
        double r116854 = r116852 * r116853;
        double r116855 = r116851 + r116854;
        double r116856 = r116850 * r116850;
        double r116857 = r116853 * r116853;
        double r116858 = r116856 + r116857;
        double r116859 = r116855 / r116858;
        return r116859;
}
↓
double f(double a, double b, double c, double d) {
        double r116860 = a;
        double r116861 = c;
        double r116862 = r116860 * r116861;
        double r116863 = b;
        double r116864 = d;
        double r116865 = r116863 * r116864;
        double r116866 = r116862 + r116865;
        double r116867 = r116861 * r116861;
        double r116868 = r116864 * r116864;
        double r116869 = r116867 + r116868;
        double r116870 = r116866 / r116869;
        double r116871 = -inf.0;
        bool r116872 = r116870 <= r116871;
        double r116873 = sqrt(r116869);
        double r116874 = r116860 / r116873;
        double r116875 = 1.4420118311925315e+300;
        bool r116876 = r116870 <= r116875;
        double r116877 = r116866 / r116873;
        double r116878 = r116877 / r116873;
        double r116879 = -1.0;
        double r116880 = r116879 * r116860;
        double r116881 = r116880 / r116873;
        double r116882 = r116876 ? r116878 : r116881;
        double r116883 = r116872 ? r116874 : r116882;
        return r116883;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.9
Target0.4
Herbie24.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < -inf.0

    1. Initial program 64.0

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt64.0

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*64.0

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

    5. Taylor expanded around inf 55.8

      \[\leadsto \frac{\color{blue}{a}}{\sqrt{c \cdot c + d \cdot d}}\]

    if -inf.0 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < 1.4420118311925315e+300

    1. Initial program 11.8

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt11.8

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*11.7

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

    if 1.4420118311925315e+300 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

    1. Initial program 63.5

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt63.5

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*63.5

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

    5. Taylor expanded around -inf 59.5

      \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{c \cdot c + d \cdot d}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification24.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} = -\infty:\\ \;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 1.4420118311925315 \cdot 10^{300}:\\ \;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))